c h a p ter 22: e l eme n tary g ra p ha lgo r i t h ms
DESCRIPTION
C h a p ter 22: E l eme n tary G ra p hA lgo r i t h ms. U ndi rected G ra p h. Information T echnology. E x ample: A city map for pedestrians (edges are streets, nodes are intersections) ABC D. E. F. G. J. I. H. K. L. M. N. D i rected Grap h(Digrap h ). - PowerPoint PPT PresentationTRANSCRIPT
Chapter 22: Elementary GraphAlgorithms
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yUndirected Graph
Example: A city map for pedestrians (edges are streets,nodes are intersections)
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yDirected Graph (Digraph)
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Example: A city map for cars (edges are one-way streets,nodes are intersections)
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Node, vertex (plural: vertices) Edge Self-loop (edge from a node to itself) Adjacent nodes (connected by an edge) Degree (#edges from or to a node) In-degree (#edges to a node) Out-degree (#edges from a node) Path (sequence of adjacent nodes)
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Adjacency-matrix representation:a 2-dimensional array A of 0/1 values, with A[i,j ] containing the number of arcsbetween nodes i and j (undirected graph),or from node i to node j (digraph)
Adjacency-list representation:a 1-dimensional array Adj of adjacency lists, with Adj [i ] containing the list of nodesadjacent to node i
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A B C D E F G H I J K L M N
A 1 1 1 1
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A B C D E F G H I J K L M N
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Note: For our convenience, each adjacency list is alphabetically ordered.
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yGraph Traversals
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by breadth-first search (BFS) by depth-first search (DFS)
BFS and DFS work on directed graphs as well as on undirected graphs(the examples below are for digraphs).
BFS and DFS assume the given graph is represented using adjacency lists.
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Breadth-first search (BFS) Depth-first search (DFS)
BFS Algorithm
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Given a source node, A. Paint the source gray. Paint other nodes white. Add the source to the initially empty first-in first-out (FIFO) queue of gray nodes.
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BFS order (black) Queue (gray)A
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Paint it black (all adjacent I:
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Dequeue head node, A. Paint its undiscovered (=white) adjacent nodes gray and enqueue them.
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nodes are discovered).G H J
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BFS order (black) Queue (gray)A BF
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BFS order (black) Queue (gray)A BFAB FC
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BFS order (black) Queue (gray)A BFAB FCABF CH
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BFS order (black) Queue (gray)A BFAB FCABF CHABFC HGJ
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BFS order (black) Queue (gray)A BFAB FCABF CHABFC HGJABFCH GJK
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BFS order (black) Queue (gray)A BFAB FCABF CHABFC HGJABFCH GJKABFCHG JK
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BFS order (black) Queue (gray)A BFAB FCABF CHABFC HGJABFCH GJKABFCHG JKABFCHGJ KILM
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G ) ( H ) ( J
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and so on
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BFS order (black) Queue (gray)A BFAB FCABF CHABFC HGJABFCH GJKABFCHG JKABFCHGJ KILMABFCHGJK ILM
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May restart from white source D
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BFS order (black) Queue (gray)A BFAB FCABF CHABFC HGJABFCH GJKABFCHG JKABFCHGJ KILMABFCHGJK ILMABFCHGJKI LMEABFCHGJKIL MEABFCHGJKILM ENABFCHGJKILME NABFCHGJKILMEN
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Analysis of BFS for graph (V,E) (without any restarts)
Aggregate analysis: O(V+E) time.20
The initialisations take Θ(V) time. Each node is enqueued (in O(1) time) and
dequeued (in O(1) time) at most once (because no node is ever repainted white), hence O(V) time for all queue operations.
Each adjacency list is scanned at most once, and their total length is Θ(E), hence O(E) time for scanning adjacency lists.
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Refinements of BFS (at no increase in time complexity)
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Store the predecessor (or parent) of each newly discovered node(initially NIL, as undiscovered = white):breadth-first tree, rooted at source s.
Store the distance (in #edges) from the source s to each newly discovered node (by adding 1 to the distance of its parent, with s being at distance 0): lengths ofshortest paths from s to all reachable nodes.
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Breadth-first tree from source A (without restart from D):
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Shortest Path
Shortest Path
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yGraph Traversals
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Breadth-first search (BFS) Depth-first search (DFS)
Depth first search Algorithm
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Paint all nodes white.DFS finish order (black) Stack (gray) Rest (white)
ABCDEFGHIJKLMN
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The first white node, A, is the first source node.
Paint the next undiscovered (=white) adjacent node gray and push it onto a stack before continuing on it (recursive case);if no such adjacent node (or: “child”) is found, then pop the top node off the stack,paint it black (all adjacent nodes discovered), and stop (base case).
DFS finish order (black) Stack (gray) Rest (white)A BCDEFGHIJKLMN
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Continue to A’s first undiscovered child, B.DFS finish order (black) Stack (gray) Rest (white)
A BCDEFGHIJKLMNBA CDEFGHIJKLMN
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Continue to B’s first undiscovered child, C.DFS finish order (black) Stack (gray) Rest (white)
A BCDEFGHIJKLMNBA CDEFGHIJKLMNCBA DEFGHIJKLMN
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Continue to C’s first undiscovered child, G.DFS finish order (black) Stack (gray) Rest (white)
A BCDEFGHIJKLMNBA CDEFGHIJKLMNCBA DEFGHIJKLMNGCBA DEFHIJKLMN
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Start at AContinue to G’s first undiscovered child, K.
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DFS finish order (black) Stack (gray) Rest (white)A BCDEFGHIJKLMNBA CDEFGHIJKLMNCBA DEFGHIJKLMNGCBA DEFHIJKLMNKGCBA DEFHIJLMN
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Continue to K’s first undiscovered child, H.DFS finish order (black) Stack (gray) Rest (white)
A BCDEFGHIJKLMNBA CDEFGHIJKLMNCBA DEFGHIJKLMNGCBA DEFHIJKLMNKGCBA DEFHIJLMNHKGCBA DEFIJLMN
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H has no undiscovered child, backtrack until we find node K, with undiscovered child J.
DFS finish order (black) Stack (gray) Rest (white)A BCDEFGHIJKLMNBA CDEFGHIJKLMNCBA DEFGHIJKLMNGCBA DEFHIJKLMNKGCBA DEFHIJLMNHKGCBA DEFIJLMN
H JKGCBA DEFILMN
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Continue to J’s first undiscovered child, I.DFS finish order (black) Stack (gray) Rest (white)
A BCDEFGHIJKLMNBA CDEFGHIJKLMNCBA DEFGHIJKLMNGCBA DEFHIJKLMNKGCBA DEFHIJLMNHKGCBA DEFIJLMN
H JKGCBA DEFILMNH IJKGCBA DEFLMN
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Continue to I’s first undiscovered child, E.DFS finish order (black) Stack (gray) Rest (white)
A BCDEFGHIJKLMNBA CDEFGHIJKLMNCBA DEFGHIJKLMNGCBA DEFHIJKLMNKGCBA DEFHIJLMNHKGCBA DEFIJLMN
H JKGCBA DEFILMNH IJKGCBA DEFLMNH EIJKGCBA DFLMN
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Continue to E’s first undiscovered child, N.DFS finish order (black) Stack (gray) Rest (white)
A BCDEFGHIJKLMNBA CDEFGHIJKLMNCBA DEFGHIJKLMNGCBA DEFHIJKLMNKGCBA DEFHIJLMNHKGCBA DEFIJLMN
H JKGCBA DEFILMNH IJKGCBA DEFLMNH EIJKGCBA DFLMNH NEIJKGCBA DFLM
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Continue to N’s first undiscovered child, M.DFS finish order (black) Stack (gray) Rest (white)
A BCDEFGHIJKLMNBA CDEFGHIJKLMNCBA DEFGHIJKLMNGCBA DEFHIJKLMNKGCBA DEFHIJLMNHKGCBA DEFIJLMN
H JKGCBA DEFILMNH IJKGCBA DEFLMNH EIJKGCBA DFLMNH NEIJKGCBA DFLMH MNEIJKGCBA DFL
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M has no undiscovered child, backtrack until we find node J, with undiscovered child L.
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DFS finish order (black) Stack (gray) Rest (white)A BCDEFGHIJKLMNBA CDEFGHIJKLMNCBA DEFGHIJKLMNGCBA DEFHIJKLMNKGCBA DEFHIJLMNHKGCBA DEFIJLMN
H JKGCBA DEFILMNH IJKGCBA DEFLMNH EIJKGCBA DFLMNH NEIJKGCBA DFLMH MNEIJKGCBA DFLHMNEI LJKGCBA DF
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L has no undiscovered child, backtrack until we find node A, with undiscovered child F.
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DFS finish order (black) Stack (gray) Rest (white)A BCDEFGHIJKLMNBA CDEFGHIJKLMNCBA DEFGHIJKLMNGCBA DEFHIJKLMNKGCBA DEFHIJLMNHKGCBA DEFIJLMN
H JKGCBA DEFILMNH IJKGCBA DEFLMNH EIJKGCBA DFLMNH NEIJKGCBA DFLMH MNEIJKGCBA DFLHMNEI LJKGCBA DFHMNEILJKGCB FA D
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F has no undiscovered child; the backtrack clears the stack.
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DFS finish order (black) Stack (gray) Rest (white)A BCDEFGHIJKLMNBA CDEFGHIJKLMNCBA DEFGHIJKLMNGCBA DEFHIJKLMNKGCBA DEFHIJLMNHKGCBA DEFIJLMN
H JKGCBA DEFILMNH IJKGCBA DEFLMNH EIJKGCBA DFLMNH NEIJKGCBA DFLMH MNEIJKGCBA DFLHMNEI LJKGCBA DFHMNEILJKGCB FA DHMNEILJKGCBFA D
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Restart from next white node, D, as source.
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DFS finish order (black) Stack (gray) Rest (white)A BCDEFGHIJKLMNBA CDEFGHIJKLMNCBA DEFGHIJKLMNGCBA DEFHIJKLMNKGCBA DEFHIJLMNHKGCBA DEFIJLMN
H JKGCBA DEFILMNH IJKGCBA DEFLMNH EIJKGCBA DFLMNH NEIJKGCBA DFLMH MNEIJKGCBA DFLHMNEI LJKGCBA DFHMNEILJKGCB FA DHMNEILJKGCBFA D
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D has no undiscovered child; backtrack; done.DFS finish order (black) Stack (gray) Rest (white)
A BCDEFGHIJKLMNBA CDEFGHIJKLMNCBA DEFGHIJKLMNGCBA DEFHIJKLMNKGCBA DEFHIJLMNHKGCBA DEFIJLMN
H JKGCBA DEFILMNH IJKGCBA DEFLMNH EIJKGCBA DFLMNH NEIJKGCBA DFLMH MNEIJKGCBA DFLHMNEI LJKGCBA DFHMNEILJKGCB FA DHMNEILJKGCBFA DHMNEILJKGCBFAD
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DFS order (of pushing): ABCGKHJIENMLFDDFS finish order (black) Stack (gray) Rest (white)
A BCDEFGHIJKLMNBA CDEFGHIJKLMNCBA DEFGHIJKLMNGCBA DEFHIJKLMNKGCBA DEFHIJLMNHKGCBA DEFIJLMN
H JKGCBA DEFILMNH IJKGCBA DEFLMNH EIJKGCBA DFLMNH NEIJKGCBA DFLMH MNEIJKGCBA DFLHMNEI LJKGCBA DFHMNEILJKGCB FA DHMNEILJKGCBFA DHMNEILJKGCBFAD
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Analysis of DFS for graph (V,E) (with restarts)
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Initialisations and restarts take Θ(V) time. Each node is visited exactly once
(because no node is ever repainted white). Each adjacency list is scanned exactly
once, and their total length is Θ(E), hence Θ(E) time for scanning adjacency lists.
Aggregate analysis: Θ(V+E) time.
It takes time linear in the size of the graph.
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Refinements of DFS (at no increase in time complexity)
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Store the predecessor (or parent) of each newly discovered node(initially NIL, as undiscovered = white):depth-first forest of depth-first trees that are rooted at the chosen source nodes.
Store the timestamps of discovering (graying) and finishing (blackening) each node: useful in many graph algorithms (for example: topological sort, see below).
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Depth-first forest (of depth-first trees)
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DFS finish order (black) Stack (gray) Rest (white)A BCDEFGHIJKLMNBA CDEFGHIJKLMNCBA DEFGHIJKLMNGCBA DEFHIJKLMNKGCBA DEFHIJLMNHKGCBA DEFIJLMN
H JKGCBA DEFILMNH IJKGCBA DEFLMNH EIJKGCBA DFLMNH NEIJKGCBA DFLMH MNEIJKGCBA DFLHMNEI LJKGCBA DFHMNEILJKGCB FA DHMNEILJKGCBFA DHMNEILJKGCBFAD
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Depth-first
forest A B
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Another depth-first forest, starting from D A B C D E
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H K LM N
Starting from L, then M, then J, then K, then
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Strongly Connected Components of a Digraph
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Strongly connected component (SCC): maximal set of nodes where there is a path from each node to each other node.
Many algorithms first divide a digraph into its SCCs, then process these SCCs separately, and finally combine the sub- solutions. (This is not divide-&-conquer!)
An undirected graph can be decomposed into its connected components.
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Strongly Connected Components of a Digraph
Strongly connected component (SCC): maximal set of nodes where there is a path from each node to each other node.
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Strongly Connected Components of a Digraph
Strongly connected component (SCC): maximal set of nodes where there is a path from each node to each other node.
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yComputing the Strongly Connected Components of a Digraph G
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1. Enumerate the nodes of G in DFS finish order, starting from any node
2. Compute the transpose GT
(that is, reverse all edges)
3. Make a DFS in GT, considering nodes in reverse finish order from original DFS
4. Each tree in this depth-first forest is a strongly connected component
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A B C D E
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DFS finish order of G:
Reverse DFS finish order of G:
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GT
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DFS finish order (black) Stack (gray) Rest (white)D AFBCGKJLIENMH
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DFS finish order (black) Stack (gray) Rest (white)D AFBCGKJLIENMH
D| A FBCGKJLIENMH
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DFS finish order (black) Stack (gray) Rest (white)D AFBCGKJLIENMH
D| A FBCGKJLIENMHD| GA FBCKJLIENMH
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DFS finish order (black) Stack (gray) Rest (white)D AFBCGKJLIENMH
D| A FBCGKJLIENMHD| GA FBCKJLIENMHD| CGA FBKJLIENMH
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DFS finish order (black) Stack (gray) Rest (white)D AFBCGKJLIENMH
D| A FBCGKJLIENMHD| GA FBCKJLIENMHD| CGA FBKJLIENMHD| BCGA FKJLIENMH
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DFS finish order (black) Stack (gray) Rest (white)D AFBCGKJLIENMH
D| A FBCGKJLIENMHD| GA FBCKJLIENMHD| CGA FBKJLIENMHD| BCGA FKJLIENMHD|BC KGA FJLIENMH
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DFS finish order (black) Stack (gray) Rest (white)D AFBCGKJLIENMH
D| A FBCGKJLIENMHD| GA FBCKJLIENMHD| CGA FBKJLIENMHD| BCGA FKJLIENMHD|BC KGA FJLIENMHD|BC HKGA FJLIENM
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H K L M N62
DFS finish order (black) Stack (gray) Rest (white)D AFBCGKJLIENMH
D| A FBCGKJLIENMHD| GA FBCKJLIENMHD| CGA FBKJLIENMHD| BCGA FKJLIENMHD|BC KGA FJLIENMHD|BC HKGA FJLIENMD|BC FHKGA JLIENM
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A B C D E
F G IJ
H K L M N63
DFS finish order (black) Stack (gray) Rest (white)D AFBCGKJLIENMH
D| A FBCGKJLIENMHD| GA FBCKJLIENMHD| CGA FBKJLIENMHD| BCGA FKJLIENMHD|BC KGA FJLIENMHD|BC HKGA FJLIENMD|BC FHKGA JLIENMD|BCFHKGA| J LIENM
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A B C D E
F G IJ
H K L M N60
DFS finish order (black) Stack (gray) Rest (white)D AFBCGKJLIENMH
D| A FBCGKJLIENMHD| GA FBCKJLIENMHD| CGA FBKJLIENMHD| BCGA FKJLIENMHD|BC KGA FJLIENMHD|BC HKGA FJLIENMD|BC FHKGA JLIENMD|BCFHKGA| J LIENMD|BCFHKGA|J| L IENM
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A B C D E
F G IJ
H K L M N65
DFS finish order (black) Stack (gray) Rest (white)D AFBCGKJLIENMH
D| A FBCGKJLIENMHD| GA FBCKJLIENMHD| CGA FBKJLIENMHD| BCGA FKJLIENMHD|BC KGA FJLIENMHD|BC HKGA FJLIENMD|BC FHKGA JLIENMD|BCFHKGA| J LIENMD|BCFHKGA|J| L IENMD|BCFHKGA|J|L I ENM
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A B C D E
F G IJ
H K L M N66
DFS finish order (black) Stack (gray) Rest (white)D AFBCGKJLIENMH
D| A FBCGKJLIENMHD| GA FBCKJLIENMHD| CGA FBKJLIENMHD| BCGA FKJLIENMHD|BC KGA FJLIENMHD|BC HKGA FJLIENMD|BC FHKGA JLIENMD|BCFHKGA| J LIENMD|BCFHKGA|J L IENMD|BCFHKGA|J|L I ENMD|BCFHKGA|J|L|I E NM
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A B C D E
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H K L M N67
DFS finish order (black) Stack (gray) Rest (white)D AFBCGKJLIENMH
D| A FBCGKJLIENMHD| GA FBCKJLIENMHD| CGA FBKJLIENMHD| BCGA FKJLIENMHD|BC KGA FJLIENMHD|BC HKGA FJLIENMD|BC FHKGA JLIENMD|BCFHKGA| J LIENMD|BCFHKGA|J L IENMD|BCFHKGA|J|L I ENMD|BCFHKGA|J|L|I E NMD|BCFHKGA|J|L|I|E N M
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A B C D E
F G IJ
H K L M N68
DFS finish order (black) Stack (gray) Rest (white)D AFBCGKJLIENMH
D| A FBCGKJLIENMHD| GA FBCKJLIENMHD| CGA FBKJLIENMHD| BCGA FKJLIENMHD|BC KGA FJLIENMHD|BC HKGA FJLIENMD|BC FHKGA JLIENMD|BCFHKGA| J LIENMD|BCFHKGA|J L IENMD|BCFHKGA|J|L I ENMD|BCFHKGA|J|L|I E NMD|BCFHKGA|J|L|I|E N MD|BCFHKGA|J|L|I|E MN
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A B C D E
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H K L M N69
DFS finish order (black) Stack (gray) Rest (white)D AFBCGKJLIENMH
D| A FBCGKJLIENMHD| GA FBCKJLIENMHD| CGA FBKJLIENMHD| BCGA FKJLIENMHD|BC KGA FJLIENMHD|BC HKGA FJLIENMD|BC FHKGA JLIENMD|BCFHKGA| J LIENMD|BCFHKGA|J L IENMD|BCFHKGA|J|L I ENMD|BCFHKGA|J|L|I E NMD|BCFHKGA|J|L|I|E N MD|BCFHKGA|J|L|I|E MND|BCFHKGA|J|L|I|E|MN
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A B C D E
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H K L M N70
DFS finish order (black) Stack (gray) Rest (white)D AFBCGKJLIENMH
D| A FBCGKJLIENMHD| GA FBCKJLIENMHD| CGA FBKJLIENMHD| BCGA FKJLIENMHD|BC KGA FJLIENMHD|BC HKGA FJLIENMD|BC FHKGA JLIENMD|BCFHKGA| J LIENMD|BCFHKGA|J L IENMD|BCFHKGA|J|L I ENMD|BCFHKGA|J|L|I E NMD|BCFHKGA|J|L|I|E N MD|BCFHKGA|J|L|I|E MND|BCFHKGA|J|L|I|E|MN
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A B C D E
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K L M N71
DFS finish order (black) Stack (gray) Rest (white)D AFBCGKJLIENMH
D| A FBCGKJLIENMHD| GA FBCKJLIENMHD| CGA FBKJLIENMHD| BCGA FKJLIENMHD|BC KGA FJLIENMHD|BC HKGA FJLIENMD|BC FHKGA JLIENMD|BCFHKGA| J LIENMD|BCFHKGA|J L IENMD|BCFHKGA|J|L I ENMD|BCFHKGA|J|L|I E NMD|BCFHKGA|J|L|I|E N MD|BCFHKGA|J|L|I|E MND|BCFHKGA|J|L|I|E|MN
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A B C F G H K
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H K L M N72
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Analysis of decompositioninto SCCs of digraphG=(V,E)
A strongly connected digraph has one SCC.73
Assume G is represented by adjacency lists:
1.DFS in G: Θ(V+E) time, including reversal
2. Compute transpose GT: Θ(V+E) time
3. DFS in GT: Θ(V+E) time
4.Output SCCs: Θ(V+E) time
Total: Θ(V+E) time
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74
A topological sort is a linear ordering of all nodes of a directed acyclic graph (DAG).
If there is a path from node ni to node nj
then ni appears before nj in the ordering.
Note on DAGs:There is no cycle (path from node to itself)There is at least one node of in-degree 0There is at least one node of out-degree 0
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75
Example: We are given courses at a university and
the prerequisite relations between. These can be modelled as a DAG. A topological sort of these courses gives
a valid sequence of taking them.
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Intro IT
PKD AD2
DArkDig Av
Algo
AI
76
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Intro IT
PKD AD2
DArkDig Av
Algo
AINode with in-degree 0
77
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Intro IT
PKD AD2
DArkDig Av
Algo
AINodes with out-degree 0
78
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TOPOLOGICAL SORT
Select node with in-degree 0
Print it out Remove it Repeat
Intro IT
PKD AD2
DArk Dig Av
Algo
AI
79
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TOPOLOGICAL SORT
Select IntroIT
Intro IT
PKD AD2
DArk Dig Av
Algo
AI
80
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TOPOLOGICAL SORT
Select IntroIT Print IntroIT
Intro IT
PKD AD2
DArk Dig Av
Algo
AI
IntroIT81
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TOPOLOGICAL SORT
Select IntroIT Print IntroIT Remove it
PKD AD2
DArk Dig Av
Algo
AI
IntroIT82
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TOPOLOGICAL SORT
Select PKD
PKD AD2
DArk Dig Av
Algo
AI
IntroIT83
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TOPOLOGICAL SORT
Select PKD Print PKDPKD AD2
DArk Dig Av
Algo
AI
IntroIT, PKD80
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TOPOLOGICAL SORT
Select PKD Print PKD Remove it
AD2
DArk Dig Av
Algo
AI
IntroIT, PKD85
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TOPOLOGICAL SORT
Select DArkDig
AD2
DArk Dig Av
Algo
AI
IntroIT, PKD86
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TOPOLOGICAL SORT
Av Algo
Select DArkDig
Print DArkDig
AD2
DArk Dig
AI
IntroIT, PKD, DArkDig87
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TOPOLOGICAL SORT
Av Algo
Select DArkDig
Print DArkDig Remove it
AD2
AI
IntroIT, PKD, DArkDig
88
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TOPOLOGICAL SORT
Av Algo
Select AI
AD2AI
IntroIT, PKD, DArkDig
89
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TOPOLOGICAL SORT
Av Algo
Select AI
Print AIAD2
AI
IntroIT, PKD, DArkDig, AI90
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TOPOLOGICAL SORT
Av Algo
Select AI
Print AI Remove it
AD2
IntroIT, PKD, DArkDig, AI91
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TOPOLOGICAL SORT
Av Algo
Select AD2
AD2
IntroIT, PKD, DArkDig, AI92
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TOPOLOGICAL SORT
Av Algo
Select AD2
Print AD2AD2
IntroIT, PKD, DArkDig, AI, AD293
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TOPOLOGICAL SORT
Av Algo
Select AD2
Print AD2
Remove it
90
IntroIT, PKD, DArkDig, AI, AD2
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TOPOLOGICAL SORT
Av Algo
Select AvAlgo
95
IntroIT, PKD, DArkDig, AI, AD2
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TOPOLOGICAL SORT
Av Algo
Select AvAlgo
Print AvAlgo
96
IntroIT, PKD, DArkDig, AI, AD2, AvAlgo
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TOPOLOGICAL SORT
97
Select AvAlgo Print AvAlgo Remove it
IntroIT, PKD, DArkDig, AI, AD2, AvAlgo
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98
A topological sort is not necessarily unique: At any step, we may have more than one node
with in-degree 0, so we have to choose one among them.
IntroIT, PKD, AI, DArkDig, AD2, AvAlgo is also a valid sequence.
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Other Algorithm andAnalysis of TopologicalSort
99
Algorithm: Compute DFS finish time of each node; as each node is finished (blackened), insert it onto the front of an initially empty linked list; return that list.
The total running time is thus Θ(V+E).